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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 Additive model of reliability of biometric systems with exponential distribution of failure probability Bihać, Bosnia and Hercegovina Zoran Ćosić(Author) jacosic@gmail.com director Statheros d.o.o. Kaštel Stari, Croatia Miroslav Bača (Author) zoran.cosic@statheros.hr professor Faculty of Organisational and Informational science Varaždin, Croatia Jasmin Ćosić (Author) miroslav.baca@foi.hr IT Section of Police Administration Ministry of Interior of Una-sana canton Abstract— Approaches for reliability analysis of biometric Data collection subsystem consists of a biometric sample, systems are subject to a review of numerous scientific papers. method of sampling, and sensors that are sampled. Signal Most of them consider issues of reliability of component software processing subsystem consists of drainage structures, quality applications. System reliability, considering technical and control and comparison of samples. Decision subsystem software part, is of crucial importance for users and for manufacturers of biometric systems. consists of the decision mechanisms and storage subsystems. In this paper, the authors developed a mathematical model to Schematisation of model described in figure 1 can be shown analyse the reliability of biometric systems, regarding the on figure 2. dependence of components with exponential distribution of failure probability. Keywords- Additive model, Biometric system, reliability, exponential distribution, UML, Figure 2 I. INTRODUCTION Schematic presentation of a biometric [1] system is a The general biometric system, according to [1] Wyman shown simplified representation of a system in Figure 1 and shows in Figure 1, consists of 5 elements which are located in all the serial configuration of system components dependence biometric systems today. II. THE DEFINITION OF THE RELIABILITY OF BIOMETRIC SYSTEMS Biometric system designers and producers are motivated to use already constructed components and modules. Component system has a high reliability expectations , no matter who is producer. Most of existing reliability models are so generally described to not consider particularity of components and modules. In this paper authors will describe methodology based on mathematical model which take account of component reliability and connections reliability also. UML methodology in describing system and his inner interaction, simplify approach for researchers. Figure 1. [1] UML [2] is also becoming standard in the process of designing and manufacturing systems so production of component Each subsystem consists of the elements that contribute to the systems gets benefits from the UML representation. overall system quality. 1 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 Assessment of [3] generic biometric system reliability in UML is given in Figure 3, which describes the use of Use Case Diagrams sequences play an important role in assessing the diagram: reliability of the system because they give information on how many components are involved in the execution of a scenario. Through sequence diagrams it is simple to count the periods of availability of components in the given scenarios as shown in figure (3). The probability of failure of components with known busy periods, can be given by the following expression: (3) Figure 3 [3] In witch is: - - probability of component failure i in the scenario j q1 and q2 represent the probability that users u1 and u2 will - - occupancy time of component i in the scenario j access the system using some of its functionality. P11 and P12 represent the probability that user u1 will use the The expression (3) applies only if the following conditions are functionality of f1 and f2, and P21 and P22 represent the same met: probability for the user u2. - Independence of failure: the probability of failure of The probability of execution of use case x, is defined by the one component does not depend on other components expression: - Regularity of failure: the probability of failure of one component is equal throughout the execution of (1) occupation period of the component m is number of users. You can also show every moment of occupancy of any component of the system considering the method to be If we are able to join a no uniform distribution to Diagram of executed at that moment in the scenario sequences in a given use case then (1) can be expressed as: If we replace with a set of method of failure probability (2) where is then equation (3) becomes: (4) Where the fj (k) - frequency of the k-th transition of sequence diagram in the j-th case. P (kj) – presents a probability of default scenarios. During system operation, components interact and exchange information. Then it is necessary to take into consideration occupation period of the component: (5) Where is θij- the probability of failure of system components and bp-busy period of the system. (6) Where is Ψlmj- the probability of failure connections between the components and - the number of interactions between system components. (7) Figure 4 Sequence diagram [3] The reliability of the system taking into account the probability of failure can be expressed as: 2 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 n2( Δt ) - number of failures in certain time interval Δt n1( t − Δt ) – the non failed number of elements at the end of (7a) the interval Δt , or until t − Δt The intensity of the element failure is calculated by the (7b) expression: 1 ) 1 III. RELIABILITY PREDICTION OF THE COMPONENTS SERIAL λEL =Θ= (11) DEPENDENCE WITH EXPONENTIAL DISTRIBUTION n Θ⋅n Based on the above assumptions [4], [5] system reliability can Where is: be calculated according to the law of exponential distribution: n- number of usable parts of the confidence interval (1 − α ) = 0, 75 (8) Θ - lower limit of confidence for the mean time between In wich are: failures Rs - System reliability λ - Intensity of system fault The total intensity of failure taking into consideration the t - Required time of reliable operation of system number of elements that are not failed in a given time is calculated by formula: Proof: (12) Where is: nEL- number of elements of subsystems that are not failed Mean time between failures MTBFs can be calculated using In a serial dependence between of all parts of the system the the expression: failure of any part of the system may cause the failure of the entire system. Failure intensity function λs in the case of a serial dependence of system elements is calculated by the expression: (13) (9) Calculation of reliability [6] of some element is based on Where is λi - failure intensity of the i-th part of the system. empirical data on the time of functioning and eventual failure of the element. Failure intensity function λs is equal to the ratio between the Problem [7], [8] becomes more complex when the information number of failures in the time-frame and the correct number of about the failure doesn't exist. In case that part worked elements in the system, until the beginning of this interval: perfectly, and information about the exploitation are available. If one assumes that the given part can apply the rule of the exponential distribution, it is possible to determine the upper (10) limit of confidence for the intensity of failure, in the cases of continuous operation or one or more failures. Where is: λs -function of failure intensity of system Lower limit of confidence for the mean time between ) Δt - failure time of an system element failures Θ , for confidence interval (1 − α ) is calculated using the formula: 3 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 probability. In accordance with this authors will define ) 2tr 2tr mathematical model for recovery system probability than Θ≥ = (14) system readiness to use. χ 2α ,2 r + 2 χ 20.25,2 Where is: REFERENCES tr – total time of system operation r – Number of elements that have failed [1] Zasnivanje otvorene ontologije odabranih segmenata biometrijske znanosti - Markus Schatten– Magistarski rad – FOI 2007 χ 2α ,2r + 2 - Random variable which has distribution [2] Modelling biometric systems in UML – Miroslav Bača, Markus Schatten, Bernardo Golenja, JIOS 2007 FOI Varaždin [3] A Bayesian Approach to Reliability Prediction and Assessment of Component IV. SPECIAL CASE OF NOT-FAILURE SYSTEM [4] Based Systems – KH. Singhy, V. Cortellessa, B. Cukic, E. Gunely,V.Bharadwaj Considering (8), (10) and (14) we obtain an expression for the [5] Department of Statistics, Lane Department of Computer Science and reliability of the whole system: Electrical Engineering West Virginia University, Proceedings of the 12th International Symposium on Software Reliability Engineering (15) (ISSREí01) 1071-9458/01 2001 IEEE [6] Simulacijsko modeliranje pouzdanosti tehničkog sustava brodskog In futher calculations: kompresora – Zoran Ćosić – Magistarski rad - 2007 [7] Teorija pouzdanosti tehničkih sistema, Vujanović Nikola, Vojnoizdavački novinski centar, Beograd 2005 (16) [8] The Impact of Error Propagation on Software Reliability Analysis of Component-based Systems – Petar Popic, Master thesis, West Vriginia 2005 Taking into account (11) we obtain an expression for the reliability of the each elements of the system: AUTHORS PROFILE Zoran Ćosić, CEO at Statheros ltd, and business consultant in business process standardization field. He received BEng degree at Faculty of nautical (17) science , Split (HR) in 1990, MSc degree at Faculty of nautical science , Split (HR) in 2007 , actually he is a PhD candidate at Faculty of informational and Organisational science Varaždin Croatia. He is a member of various professional societies and program V. CONCLUSION AND FURTHER RESEARCH committee members. He is author or co- author more than 20 scientific and professional papers. His main The reliability of technical systems is the subject for many fields of interest are: Informational security, biometrics and privacy, research scientists, according to that analysis are available in business process reingeenering, different models of reliability of biometric systems that in Jasmin Ćosić has received his BE (Economics) degree from University of most cases take into account the software as one of the Bihać, B&H in 1997. He completed his study in Information Technology field (dipl.ing.Information Technlogy) in Mostar, University of Džemal components of the same system. This study developed Bijedić, B&H. Currently he is PhD candidate in Faculty of Organization mathematical model of a generic biometric system that and Informatics in Varaždin, University of Zagreb, Croatia. He is consider user, hardware and software influence and assumes working in Ministry of the Interior of Una-sana canton, B&H. He is a serial dependence of the components and the exponential ICT Expert Witness, and is a member of Association of Informatics of B&H, Member of IEEE and ACM. His areas of interests are Digital distribution of failure probability of system components. Forensic, Computer Crime, Information Security and DBM Systems. He Scientific contribution is expressed through mathematical has presented and published over 20 conference proceedings and journal model definition of biometric system reliability prediction articles in his research area within special case where is not possible to get failure data or Miroslav Bača is currently an Associate professor, University of Zagreb, in the case of perfectly working system. The results of Faculty of Organization and Informatics. He is a member of various professional societies and program calculations can prevent real failure events by defining committee members, and he is reviewer of several international preventive maintenance. journals and conferences. He is also the head of the Biometrics centre in This mathematical model allows the prediction of system Varaždin, Croatia. He is author or co- author more than 70 scientific and professional papers and two reliability in the early fase of projecting of its components . books. His main research fields are computer forensics, biometrics and The subject of further research will be to create an privacy professor at Faculty of informational and Organisational science integrated mathematical model of reliability of complex Varaždin Croatia systems that considers the parallel and combined dependency of system components with different distributions of failure 4 http://sites.google.com/site/ijcsis/ ISSN 1947-5500